A177975 - OEIS

%I #20 Jun 12 2021 09:05:25

%S 1,0,1,0,1,1,0,2,2,1,0,2,5,3,1,0,4,7,9,4,1,0,2,14,16,14,5,1,0,6,13,34,

%T 30,20,6,1,0,4,27,43,69,50,27,7,1,0,6,26,83,107,125,77,35,8,1,0,4,39,

%U 100,209,226,209,112,44,9,1,0,10,38,155,295,461,428,329,156,54,10,1

%N Square array T(n,k) read by antidiagonals up. Each column is the first column in the matrix inverse of a triangular matrix that is the k-th differences of A051731 in the column direction.

%C The first column in this array is the first column in A134540 which is the matrix inverse of A177978. The second column is the first column in A159905. The rows in this array are described by both binomial expressions and closed form polynomials.

%H Seiichi Manyama, <a href="/A177975/b177975.txt">Antidiagonals n = 1..140, flattened</a>

%F From _Seiichi Manyama_, Jun 12 2021: (Start)

%F G.f. of column k: Sum_{j>=1} mu(j) * x^j/(1 - x^j)^k.

%F T(n,k) = Sum_{d|n} mu(n/d) * binomial(d+k-2,d-1). (End)

%e Table begins:

%e   1..1...1...1....1.....1.....1......1......1.......1.......1

%e   0..1...2...3....4.....5.....6......7......8.......9......10

%e   0..2...5...9...14....20....27.....35.....44......54......65

%e   0..2...7..16...30....50....77....112....156.....210.....275

%e   0..4..14..34...69...125...209....329....494.....714....1000

%e   0..2..13..43..107...226...428....749...1234....1938....2927

%e   0..6..27..83..209...461...923...1715...3002....5004....8007

%e   0..4..26.100..295...736..1632...3312...6270...11220...19162

%e   0..6..39.155..480..1266..2975...6399..12825...24255...43692

%e   0..4..38.182..641..1871..4789..11103..23807...47896...91367

%e   0.10..65.285.1000..3002..8007..19447..43757...92377..184755

%e   0..4..50.292.1209..4066.11837..30920..74139..165748..349438

%e   0.12..90.454.1819..6187.18563..50387.125969..293929..646645

%e   0..6..75.473.2166..8101.26202..75797.200479..492406.1136048

%e   0..8.100.636.2976.11482.38523.115915.319231..816421.1960190

%e   0..8.100.696.3546.14712.52548.167112.483879.1296064.3249312

%o (PARI) T(n, k) = sumdiv(n, d, moebius(n/d)*binomial(d+k-2, d-1)); \\ _Seiichi Manyama_, Jun 12 2021

%Y Column k=1..5 gives A063524, A000010, A007438, A117108, A117109.

%Y Main diagonal gives A332470.

%Y Cf. A177976, A177977.

%K nonn,tabl

%O 1,8

%A _Mats Granvik_, May 16 2010